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The Mori program and Non-Fano toric Homological Mirror Symmetry


Authors: Matthew Ballard, Colin Diemer, David Favero, Ludmil Katzarkov and Gabriel Kerr
Journal: Trans. Amer. Math. Soc. 367 (2015), 8933-8974
MSC (2010): Primary 14J33, 53D37; Secondary 18E30, 14T05, 14L24
Published electronically: March 13, 2015
MathSciNet review: 3403076
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Abstract: In the case of toric varieties, we continue the pursuit of Kontsevich's fundamental insight, Homological Mirror Symmetry, by unifying it with the Mori program. We give a refined conjectural version of Homological Mirror Symmetry relating semi-orthogonal decompositions of the $ B$-model on toric varieties to semi-orthogonal decompositions on the $ A$-model on the mirror Landau-Ginzburg models.

As evidence, we prove a new case of Homological Mirror Symmetry for a toric surface whose anticanonical bundle is not nef, namely a certain blow-up of $ \mathbb{P}^2$ at three infinitesimally near points.


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Additional Information

Matthew Ballard
Affiliation: Department of Mathematics, University of Wisconsin-Madison, Madison, Wisconsin 53706 — and — Fakultät für Mathematik, Universität Wien, Wien, Österreich
Email: ballard@math.wisc.edu

Colin Diemer
Affiliation: Department of Mathematics, University of Miami, Coral Gables, Florida 33124 — and — Fakultät für Mathematik, Universität Wien, Wien, Österreich
Email: cdiemer@gmail.com

David Favero
Affiliation: Fakultät für Mathematik, Universität Wien, Wien, Österreich
Email: favero@gmail.com

Ludmil Katzarkov
Affiliation: Department of Mathematics, University of Miami, Coral Gables, Florida 33124 — and — Fakultät für Mathematik, Universität Wien, Wien, Österreich
Email: lkatzark@math.uci.edu

Gabriel Kerr
Affiliation: Department of Mathematics, University of Miami, Coral Gables, Florida 33124
Email: gabriel.d.kerr@gmail.com

DOI: https://doi.org/10.1090/S0002-9947-2015-06541-6
Received by editor(s): August 1, 2013
Received by editor(s) in revised form: June 9, 2014
Published electronically: March 13, 2015
Additional Notes: The authors were funded by NSF DMS 0854977 FRG, NSF DMS 0600800, NSF DMS 0652633 FRG, NSF DMS 0854977, NSF DMS 0901330, FWF P 24572 N25, by FWF P20778 and by an ERC Grant. The first author was funded, in addition, by NSF DMS 0838210 RTG
Article copyright: © Copyright 2015 American Mathematical Society